(additive functor)
A covariant functor \(F: \mca \to \mcb\) between abelian categories is additive if the induced map \[ \Hom_{\mca}(A, A') \to \Hom_{\mcb}(F(A), F(A')) \] is a homomorphism of abelian groups, that is, \(F(f+g) = F(f) + F(g)\) for all \(f, g \in \Hom(A,A')\).
An additive functor sends zero morphisms to zero morphisms, so it carries a complex to a complex: \(F(\delta^{i+1}) \circ F(\delta^i) = F(\delta^{i+1}\circ \delta^i) = F(0) = 0\). If \(f-g=\delta h+h\delta\) is a homotopy of morphisms of complexes, then additivity gives \(F(f)-F(g)=F(\delta)F(h)+F(h)F(\delta)\), so \(F\) carries homotopic morphisms to homotopic morphisms. Since any two injective resolutions of an object are homotopy equivalent, \(R^iF\) is well defined up to canonical isomorphism for additive \(F\). An additive functor also preserves finite biproducts.
Dependencies
D-DEFABCATAbelian categoriesuses